Process Algebra (2IF45) Basic notions of Equational theory

Process Algebra (2IF45)

Basic notions of Equational theory

Dr. Suzana Andova

2

Deduction system for “Counting down”


Language (signature)

0 “zero”

s(_) “successor function”

a(_, _) “addition”

m(_, _) “multiplication”

s(s(0))

s(0)

0

1

1

0

1

1

y y’

a(x,y) a(x, y’)

s(x) x1

1

1 x x’ , y a(x,y) x’

x, y a(x,y)

Deduction rules

Terms s(s(0))

a(s(0),s(0))

a(s(0),0)

0

1

1

Terms a(s(0),s(0))

3

Deduction system for “Counting down”


Language (signature)

0 “zero”



m(_, _) “multiplication”

s(s(0))

s(0)

0

1

1

0

1

1

y y’

a(x,y) a(x, y’)

s(x) x1

1

1 x x’ , y a(x,y) x’

x, y a(x,y)

Deduction rules

Terms s(s(0))

a(s(0),s(0))

a(s(0),0)

0

1

1

Terms a(s(0),s(0))

specification

dynamic behaviour

equivalent behaviours

equal specification

4

Deduction system vs. Equational theory


Language (signature) 0 “zero”



0

1

1 y y’ a(x,y) a(x, y’)

s(x) x1

1

1 x x’ , y a(x,y) x’

x, y a(x,y)

Deduction rules

Terms e.g. a(s(0),s(0)), 0,. a(x,y)

s(s(0))

s(0)

0

1

1

a(s(0),s(0))

a(s(0),0)

0

1

1

Equivalence relation

Equalities on terms

…├ a(s(0),s(0)) = s(s(0))

5 Process Algebra (2IF45)

componts’ specifications

Equational theory

the whole system specification

the state space

reductionon specification

reductionon specification

reductionon LTSs

composition by axiom

SOS rules

• simpler• smaller• in a particular form (basic)• …

6

Example: Towards Equational theory for Arithmetic of Natural numbers P(eano) A(rith.)


a(0,s(0))

0

1

s(0)

0

1

So we need a set of axioms from which we can derive:

PA ├ a(0,s(0)) = …..= s(0) and also PA ├ a(0,0) = …..= 0 and also …..

a(0,0) 0

1. Maybe axioms

(Ax11) a(0,s(0)) = s(0)(Ax12) a(0,0) = 0

2. Maybe axioms

(Ax21) a(0,s(x)) = s(x)(Ax22) a(0,0) = 0

3. Maybe axioms

(Ax31) a(0,y) = y

7

Example: Towards Equational theory for Arithmetic of Natural numbers PA (cont.)


a(0,s(0))

0

1

s(a(0,0))

0

1

So we need a set of axioms from which we can derive:

PA ├ a(0,s(0)) = …..= s(a(0,0)) and also PA ├ a(s(0),s(0)) = …..= s(s(0)) …..

1’. Maybe axioms

(Ax11) a(s(0),s(0)) = s(s(0))(Ax12) a(0,s(0)) = s(0)

2’. Maybe axioms

(Ax21) a(s(x),s(y)) = s(s(a(x,y)))

3’. Maybe axioms

(Ax31) a(x,s(y)) = s(a(x,y))(Ax32) a(s(x),y) = s(a(x,y))

8



(Ax11) a(s(0),s(0)) = s(s(0))(Ax12) a(0,s(0)) = s(0)

(Ax21) a(s(x),s(y)) = s(s(a(x,y)))


1. Maybe axioms

(Ax11) a(0,s(0)) = s(0)(Ax12) a(0,0) = 0

2. Maybe axioms

(Ax21) a(0,s(x)) = s(x)(Ax22) a(0,0) = 0

3. Maybe axioms

(Ax31) a(0,y) = ynot generalinfinite

9



(Ax11) a(s(0),s(0)) = s(s(0))(Ax12) a(0,s(0)) = s(0)

(Ax21) a(s(x),s(y)) = s(s(a(x,y)))


1. Maybe axioms

(Ax11) a(0,s(0)) = s(0)(Ax12) a(0,0) = 0

2. Maybe axioms

(Ax21) a(0,s(x)) = s(x)(Ax22) a(0,0) = 0

3. Maybe axioms


inst

ance

of

10



(Ax11) a(s(0),s(0)) = s(s(0))(Ax12) a(0,s(0)) = s(0)

(Ax21) a(s(x),s(y)) = s(s(a(x,y)))


1. Maybe axioms

(Ax11) a(0,s(0)) = s(0)(Ax12) a(0,0) = 0

3. Maybe axioms


3. Maybe axioms

(Ax31) a(0,y) = y

? but can we derive here

a(s(x),0) = s(x) ?

11

Example: Equational theory for Arithmetic of Natural numbers PA (cont.)



3. Maybe axioms

(Ax31) a(0,y) = y

Axioms in PA

(PA1) a(x,0) = x(PA2) a(x,s(y)) = s(a(x,y))

12

Equational Theory


Question: • What did we have to take into account during our quest for the right set of axioms?

Answer: • Consistency between

bisimulation and

derivation in PA using the axioms

13

Consistency


Questions: • How do you understand “consistency” between the deduction system (semantics) and the equational theory?

Answer: • Consistency is two-directional

1. Everything that I am able to derive equal using the axioms must be bisimilar, namely

if PA ├ t = r then t r

2. Everything that I can show bisimilar, I have to be able to derive equial using the axioms, namely

if t r then PA ├ t = r

soundness

completeness

14

Consistency


Questions: • How do you understand “consistency” between the deduction system (semantics) and the equational theory?

Answer: • Consistency is two-directional

1. Everything that I am able to derive equal using the axiomsmust be bisimilar, namely


2. Every closed terms that I can show bisimilar, I have to be able to derive equial using the axioms, namely

if t r then PA ├ t = r, for t and r closed terms

soundness

ground completeness

15

Prerequisites for soundness of axioms


Questions:• Are we on a safe ground now?

Soundness: Everything that I am able to derive equal using the axioms must be bisimilar, namely


16

Derivation in equational theory


Questions:• What is allowed (can be used) in derivation?

PA ├ t = r

1.Axioms (as ‘basic’ equalities)

2.Properties of = such as• reflexivity PA ├ t = t • symmetry if PA ├ t = r then PA ├ r = t • transitivity if PA ├ t = r and PA ├ r = u then PA ├ t = u

3.Substitution (substituting variables by terms)e.g. from PA ├ a(x,y) = a(y,x) follows PA ├ a(s(0),0) = a(0, s(0)) where s(0)/x and 0/y

4.Context rule

17

Derivation in equational theory


Questions:• What is allowed (can be used) in derivation?

PA ├ t = r

4.Context rulee.g. if PA ├ t1 = r1 and PA ├ t2 = r2

then PA├ s(t1) = s(r1) and also PA├ a(t1 ,t2) = a(r1, r2) …

‘Mapping’ the context rule back to semantics and

e.g. if PA ├ t1 r1 and PA ├ t2 r2

then PA├ s(t1) s(r1)

and also PA├ a(t1 ,t2) a(r1, r2) …

18

Prerequisites for soundness


‘Mapping’ the context rule back to semantics and

e.g. if t1 r1 and t2 r2

then s(t1) s(r1)

and also a(t1 ,t2) a(r1, r2) …

a(0,s(0))

0

1

s(a(0,0))

0

1

Sincethen

a(0,s(0))

0

1

s(a(0,0))

0

1

s(a(0,s(0)))

1

s(s(a(0,0)))

1

congruence

19

Prerequisites for soundnessCongruence relation


Congruence relation1. it is equivalence relation

• reflexive• symmetric• transitive

2. it is preserved by any context C[ _ ]

if t1 r1 then C[t1] C[r1]

Examples: Bisimilarity is congruence on the set of all terms

20 Process Algebra (2IF45)

Prerequisites for soundnessCongruence relation

!tea !coffee

?coin

!tea !coffee

?coin ?coin

complete trace equivalent LTSs {?coin !coffee, ?coin !tea}

put in the context “communicate with the User”

!coin ?coffee

coin

coffee coffee

coin coin

NOT complete trace equivalent LTSs {coin coffee} vs. {coin coffee, coin}

21

Towards ground completeness


Questions:• Is the set of axioms in PA (PA1) a(x,0) = x

(PA2) a(x,s(y)) = s(a(x,y))ground complete wrt the derivation rules and ?

Every closed terms that I can show bisimilar, I have to be able to derive equal using the axioms, namely

if t r then PA ├ t = r, for t and r closed terms

• What about PA ├ a(t,r) = a(r,t)? Can we derive this equality from the two axioms, for closed terms t and r?

22



Working with closed terms: 1. Basic terms

• defined in a structural way, such as: I. 0 is a basic term in PAII. if t is a basic term in PA then s(t) is also a basic term in

PA• Thus, in PA basic terms are 0, s(0), s(s(0)), ….

2. Elimination of ‘other’ operators from the signature• In PA every closed term can be rewritten to a basic term, e.g.

PA ├ a(s(0),a(s(0),0)) = s(s(0))

• the elimination proof goes by structural induction on the structure on closed terms

3.Any property on closed terms can be proven by structural induction on basic terms only instead on all closed terms

23



Property1: For any closed term t, PA ├ a(0, t) = t.

Proof: By structural induction on basic terms (knowing the elimination property)

Basic step t 0. Directly from PA1

Inductive step t s(t’) for some basic term t’. DerivePA ├ a(0, t) = a(0, s(t’)) = s(a(0,t’)) = s(t’) = t

PA2 by induction

24



Property2: For any closed term t and r, PA ├ a(s(t), r) = a(t, s(r)).

Proof: By structural induction on basic terms (knowing the elimination property)

Basic step r 0. PA ├ a(s(t), r) = a(s(t), 0) = s(t) by axiom PA1PA ├ a(t, s(r)) = a(t, s(0)) = s(a(t, 0)) = s(t) by PA2 and PA1

Inductive step r s(r’) for some basic term r’. DerivePA ├ a(s(t), r) = a(s(t), s(r’)) = s( a(s(t), r’) ) = s( a(t, s(r’)) )

= a( t, s(s(r’)) ) = a(t, s(r))

PA2 by induction

PA2

25



Property3: For any closed term t and r, PA ├ a(t, r) = a(r, t).

Proof: Home Work


Basic Process Algebra

Dr. Suzana Andova

Documents

Process Algebra (2IF45) Basic notions of Equational theory